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Theorem iota4an 5077
Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4an  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )

Proof of Theorem iota4an
StepHypRef Expression
1 iota4 5076 . 2  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ph  /\  ps ) )
2 euiotaex 5074 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  ( iota
x ( ph  /\  ps ) )  e.  _V )
3 simpl 108 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
43sbcth 2895 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
52, 4syl 14 . . 3  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
6 sbcimg 2922 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  ( [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) ) )
72, 6syl 14 . . 3  |-  ( E! x ( ph  /\  ps )  ->  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ( ph  /\ 
ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) ) )
85, 7mpbid 146 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph ) )
91, 8mpd 13 1  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   E!weu 1977   _Vcvv 2660   [.wsbc 2882   iotacio 5056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-sn 3503  df-pr 3504  df-uni 3707  df-iota 5058
This theorem is referenced by: (None)
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